It cannot be the case that ##E[Hw(Hw)^H]=THH^H## because the LHS is not a random variable, whereas the RHS is.

We can write ##E[Hw(Hw)^H]=E[Hww^HH^H]## but what, if anything, can be done from there depends on what we know about ##w##. Is ##w## a random variable? If so, do we know anything about its distribution other than that each of its components has zero mean?